A goldsmith desires to test the purity of a gold ornament suspected to the mixed with copper. The ornament weighs `0.25 kg` in air and is observed to displace `0.015` litre of water when immersed in it. Densities of gold and copper with respect to water are, respectively, `19.3` and `8.9`. The approximate percentage of copper in the ornament is

Options

A. `5%`
B. `10%`
C. `15%`
D. `25%`

Explanation

To determine the percentage of copper in the gold ornament, we will follow these steps:

### Step 1: Calculate the Volume of the Ornament
The volume of water displaced by the ornament is given as `0.015 liters`. Since `1 liter = 1000 cm³`, we can convert this to cubic centimeters:
\[ V = 0.015 \, \text{liters} \times 1000 \, \text{cm}^3/\text{liter} = 15 \, \text{cm}^3 \]

### Step 2: Calculate the Density of the Ornament
The density of the ornament can be calculated using the formula:
\[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \]
The mass of the ornament is given as `0.25 kg`, which is equivalent to `250 grams` (since `1 kg = 1000 grams`):
\[ \text{Density}_{\text{ornament}} = \frac{250 \, \text{g}}{15 \, \text{cm}^3} \approx 16.67 \, \text{g/cm}^3 \]

### Step 3: Set Up the Equation for the Mixture
Let \( m \) be the mass of copper in grams. The mass of gold will then be \( 250 - m \) grams. The densities of gold and copper are given as:
- Density of gold: \( 19.3 \, \text{g/cm}^3 \)
- Density of copper: \( 8.9 \, \text{g/cm}^3 \)

The total volume of the ornament can be expressed as:
\[ \text{Volume}_{\text{gold}} + \text{Volume}_{\text{copper}} = 15 \, \text{cm}^3 \]
Using the formula for volume:
\[ \frac{250 - m}{19.3} + \frac{m}{8.9} = 15 \]

### Step 4: Solve for \( m \)
Multiply through by the least common multiple of the denominators (which is \( 19.3 \times 8.9 \)):
\[ (250 - m) \cdot 8.9 + m \cdot 19.3 = 15 \cdot (19.3 \cdot 8.9) \]
Expanding this gives:
\[ 8.9 \cdot 250 - 8.9m + 19.3m = 15 \cdot (19.3 \cdot 8.9) \]
Combining like terms:
\[ (19.3 - 8.9)m = 15 \cdot (19.3 \cdot 8.9) - 8.9 \cdot 250 \]
Calculating the right-hand side:
\[ 15 \cdot (19.3 \cdot 8.9) - 8.9 \cdot 250 = 15 \cdot 171.77 - 2225 = 2576.55 - 2225 = 351.55 \]
Thus, we have:
\[ 10.4m = 351.55 \]
Solving for \( m \):
\[ m \approx \frac{351.55}{10.4} \approx 33.8 \, \text{grams} \]

### Step 5: Calculate the Percentage of Copper
To find the percentage of copper in the ornament:
\[ \text{Percentage of copper} = \left( \frac{m}{\text{Total mass}} \right) \times 100 = \left( \frac{33.8}{250} \right) \times 100 \approx 13.52\% \]

### Final Result
The approximate percentage of copper in the ornament is **13.52%**.

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